Problem: Solve for $x$, ignoring any extraneous solutions: $\dfrac{x^2 + 14x}{x - 6} = \dfrac{29x - 54}{x - 6}$
Explanation: Multiply both sides by $x - 6$ $ \dfrac{x^2 + 14x}{x - 6} (x - 6) = \dfrac{29x - 54}{x - 6} (x - 6)$ $ x^2 + 14x = 29x - 54$ Subtract $29x - 54$ from both sides: $ x^2 + 14x - (29x - 54) = 29x - 54 - (29x - 54)$ $ x^2 + 14x - 29x + 54 = 0$ $ x^2 - 15x + 54 = 0$ Factor the expression: $ (x - 6)(x - 9) = 0$ Therefore $x = 6$ or $x = 9$ However, the original expression is undefined when $x = 6$. Therefore, the only solution is $x = 9$.